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					August 01 Umbono Fund Managers
Member of Umbono Financial Services Group

Quantitative Research Quantitative Research

Understanding Tracking Error

Quantitative Research Craig Chambers August 2001 Umbono Fund Managers 1

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Quantitative Research

Umbono Fund Management Team

Chief Executive Officer

: Tendai Musikavanhu

27-11-484 5005 082 371 5037 27-11-484 5005 083 307 9549 27-11-484 5005 082 447 8190 27-11-484 5005 082 905 9863

tmusika@umbono.co.za

Chief Investment Officer

: Craig Chambers

cchambers@umbono.co.za

Quantitative Analyst

: Dharma Laloobhai

dlaloo@umbono.co.za

Team Assistant

: Ziyanda Mbuyazi

zmbuyazi@umbono.co.za

This document is strictly confidential and is intended solely for the person or organisation to whom it is prepared for. It may contain privileged and confidential information and , if you are not the intended recipient, you may not copy or distribute it or take action in reliance on it. Any unauthorised interception of this document is illegal. Save for bona fide company matters, neither Umbono Fund Managers, nor any other person, juristic or natural within the Umbono and Ignision Groups of companies, accepts any responsibility for the opinions expressed in this document.

Umbono Fund Managers 2nd Floor, West Wing Oakhurst building 13 St. Andrews Road Parktown,Johannesburg Postnet suite #201 Private Bag x30500 Houghton 2041 Registered no : 2000/028675/07 Telephone +27 11 484 5005 Fax +27 11 484 5004

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Indexation: Defining and Dissecting “Tracking Error”
Introduction
The new millennium may not have ushered in many Y2K viruses but in the world of Fund Managers and Asset Consultants, the use of tracking error analysis has become far more pervasive over the past year and a half. This research report addresses the definition, application and sources of tracking error (TE) as well as provides a practical breakdown of factors contributing to TE costs.

Definition
Standard Deviation is not an absolute measure

Standard Deviation measures the spread of values around a mean value. Tracking error is defined as the standard deviation of the difference in returns between a tracker fund and the relevant benchmark index, measured over a certain period of time. What one must understand is that this is a statistical measure which carries probabilistic information. By this we mean that TE describes the distribution characteristics of the return differences, as apposed to an absolute measurement of return differences. It is because of this “probabilistic” nature of the TE measurement that it can be used to predict future expected return differences. This measurement utilises one standard deviation which implies that 67% of the time, the fund’s return difference, relative to it’s benchmark, will fall within the TE. (Please note that no differentiation is made between whether this difference will be from out performance or under performance) The above definition is the most widely used, however on occasion, TE has been alternatively defined as the cumulative difference in returns between the fund and it’s benchmark. This is an incorrect measurement as it is based purely on historical returns and cannot be used to predict future TE. This measurement will also appear to be far greater as the compounding effect will accentuate the difference, especially over long periods.

TE utilises one Standard Deviation

Using cumulative difference in returns is an incorrect measure

Standard Deviation Interpretation Problem
Large, once-off return differences will skew the S.D.

The accuracy of the standard deviation calculation increases as the number of return difference data points increases. Most calculations will incorporate 36 monthly, rolling data points, all of which are equally weighted. Therefore, an unusually large, once-off return difference will skew the standard deviation. For example, a tracker fund holding a small percentage of cash during a market correction may cause a once-off, return difference ‘spike’. This would detrimentally impact the TE and would remain in the rolling calculation for 36 months. Global research is currently being done to determine the appropriate weighting to be given to this type of event.

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Application
TE is used to describe the “tightness” of historical tracker portfolio performances around the actual benchmark index returns. This gives an indication of the ‘volatility’ of the tracker fund’s returns relative to the benchmark.
TE derived from a multi-factor model is more accurate

As previously discussed, TE is ‘probabilistic’ in nature and is therefore a relatively sound predictor of a fund’s future return differences. However, a TE derived from a multi-factor model (e.g. Barra) has even better predictive capabilities. A multifactor model is essentially a simulation tool that predicts portfolio behaviour based on how stocks have moved historically, relative to one another (co-variance). Barra’s ‘black-box’ consists of a very sophisticated co-variance matrix that has been proven over time to be an accurate predicator of a fund’s future TE

Sources of Tracking Error
The sources or components of TE are decomposed and classified into two main categories:

• Non-controllable TE

• The fund manager has no control over this component of the TE. These

slippage costs are relatively certain and are well defined

• Partially controllable TE • To some extent the fund manager does have control over this component.
However the eventual costs are uncertain and are generally not well defined

TE Source

Non controlled TE
Fees charged

Partially Controlled TE

Fees
Management fees Custodian fees

None

Cash Movements
Deposits, withdrawals and cash dividends will require shares to be bought and sold to re-balance a fund’s exposure

Transaction Costs: •Brokerage •MST

Impact costs from: •Market timing •Agency vs portfolio trade decision • VWAP vs Closing price decision •Broker monitoring •etc Impact costs from: •Market timing •Tranche policy (eg. 1/3 before inclusion, 1/3 on the day and 1/3 after inclusion)

Benchmark Index Changes
Constituent exclusions or inclusions Corporate actions can also lead to changes in the relative weighting of index constituents

Transaction Costs

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TE Source Performance Calculations
Return calculations generally assume a simplistic approach to dividends and other cash movements (timing). For example, ‘Money Weighted Return’ vs ‘Time Weighted Return’

Non controlled TE
Prescribed Calculation method

Partially Controlled TE
Preferred Calculation Method

Rebalancing Fund After optimisation
Non-replicating tracker funds imply periodic re-balancing actions, according to a specific optimising algorithm

None

Transaction costs from: •Basket trade (lower TE, higher trading costs) •Fewer individual trades (higher TE, lower trading costs)

Basis For Tracking Error Guarantees
TE will ensure diligent attention from Fund Managers

A TE guarantee serves the purpose of insuring a minimum deviation of investment returns from a specified benchmark index, thus ensuring diligent attention from the Fund Manager. The TE guarantee could be addressed as follows: • Exclude the “non-controllable TE” component (slippage costs) from the TE guarantee, OR • Make a pre-determined provision for the “non-controllable TE” by increasing the total TE guarantee

Numeric decomposition of TE
Replicated vs Optimised Portfolios

The following breakdown is an estimation that has been derived from the practical management of tracker funds. To ensure that the estimates are realistic, a scenario analysis approach has been used. Please note that one must always distinguish between Replicated Portfolios ( fund is identical to benchmark) and Optimised Portfolios ( fund consists of less counters than benchmark). Optimised Tracker Portfolio

Good TE
Non-controlled TE Partially controlled TE TOTAL 0.14% 0.78% 0.93%

Normal TE
0.31% 1.11% 1.42%

Bad TE
0.44% 1.70% 2.14%

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Replicated Tracker Portfolio

Good TE
Non-controlled TE Partially controlled TE TOTAL 0.15% 0.04% 0.18%

Normal TE
0.28% 0.14% 0.42%

Bad TE
0.42% 0.42% 0.84%

Factors that impact TE
Different factors will have a high, normal or low impact upon TE

This section is an attempt to give the reader a far more practical understanding of when different factors (measured as a % of total fund) will have a low, normal or high impact upon a fund’s TE. For example, if the total ‘cash inflows’ (1st factor in table below), were approximately 10% of the total portfolio, over a one year period, then the impact upon TE would be low. This would be as a result of low re-balancing costs. However, as cash inflows increase, so does the impact upon TE. As with the previous section, these percentages have been based upon actual portfolios, however, it must be stated that they are merely estimates.

Factors
Cash Inflows Cash Outflows Index Inclusions Index Exclusions Corporate Actions Optomisation Turnover Dividend Yield Brokerage Costs Market Timing Costs Performance Calculations *

Low TE Impact
10% 10% 6% 5% 5% 5% 2% 0.12% 0.20% 0.05%

Normal TE Impact
25% 15% 10% 8% 10% 15% 2.75% 0.15% 0.30% 0.07%

High TE Impact
30% 30% 16% 13% 15% 20% 4% 0.17% 0.50% 0.09%

* Performance calculation differences incurred due to dividend yield mismatchs

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Conclusion
TE is only an effective ‘policing’ tool, if it is properly understood

The future reliance placed upon TE by Fund Managers and Asset Consultants is predicted to increase. Over the next 5 to 10 years, the asset management industry will probably undergo a number of structural changes. These changes include: •Allocation of specialised mandates, with clearly defined benchmarks •Core/Satellite approach, where active and passive mandates will co-exist •Revised Regulation 28-draft: “Trustees will be required to adopt an investment strategy specific to the fund. Trustees must seek professional advice.”
Source: Institute of Retirement Funds (Jan 2001)

All of the above changes will require more “policing” of Fund Managers by Trustees, Asset Consultants and Multi-Managers. TE is an effective “policing” measure, however, an understanding of this ‘tool’ is imperative, if it is to be truly effective.

References
•Barra International •Deutsche Securities •OMAM Quant Team

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